Quantum networks for entanglement concentration Primary tabs

Student: Phillip R. Kaye

Supervisor: Prof. Michele Mosca

Background

Quantum computation and quantum cryptography have both been hot topics in theoretical computer science in recent years. Quantum algorithms offer significant advantage over any known classical algorithm in the solution of certain important computing problems, and quantum cryptography opens the possibility of perfectly secure transmission of information.

I investigate a particular type of quantum algorithm that has an important application in quantum cryptography. To fully appreciate of the nature of this investigation, familiarity with the fundamental principles of quantum computing and quantum cryptography is necessary, and the reader is referred to the homepage of the Centre for Quantum Computation, University of Oxford or to "A brief overview of quantum computing".

The fundamental unit of information in the classical setting is the bit. Physically, a bit is a two-state system whose states are assigned the values ‘0’ and ‘1’. Some quantity of information can be represented by a collection of bits taken together. We sometimes call such a collection of bits a register. A register may have n individual bits, and the state of the register at any time is completely determined by the states of its n constituent bits. For example, the state of an 8-bit register may be 11010011.

The fundamental unit of quantum information is a quantum bit, or qubit. The state of a qubit is described in terms of a state vector. This vector lives in a two-dimensional complex vector space called a Hilbert space. However, where a classical bit can only be in the state 0, or the state 1, a qubit can be in a state which is any linear combination of it's basis states, 0 and 1. For a brief mathematical introduction to the concept of a qubit, please see the quobit page.

The most marked departure from the classical case occurs when we consider “quantum registers”. Whereas the state of a classical register is completely determined by the states of its constituent bits, this is not true of quantum registers and their constituent qubits. In fact, a quantum register can exist in states for which it is impossible, even in principle, to describe the state of any individual qubit in the register. This remarkable fact is due to a phenomenon called entanglement, which is unique to quantum systems. For more information on entanglement, visit the entanglement quantum states page.

Project description

In quantum cryptography, an essential ingredient is the sharing of an EPR pair (named after Einstein, Podolsky and Rosen who wrote a famous paper discussing such particle pairs in 1935) by Alice (the ‘sender’ of some information) and Bob (the ‘receiver’). When Alice and Bob share an EPR pair, they are able to perform a process known as quantum teleportation. It has been shown by Buhrman, Cleve and Wigderson that using certain protocols involving the sharing of EPR pairs, information exchange can be achieved using fewer bits than could be achieved using a purely classical channel (the advantage is square-root in the number of bits).

Implementing these protocols in practice, however, poses a major problem, since sharing EPR pairs in reality is not an easy task. When transmitting one portion of an EPR pair over free-space or an optical fibre, we always have some degradation, or corruption, and the result is never a perfect EPR pair, and at the end of the day, Alice and Bob will share something that is slightly less entangled.

Fortunately, however, this is not the end of the story. From, say, 100 such partially entangled states shared by Alice and Bob (whom are geographically separated), it is possible to distil some perfect EPR pairs with Alice and Bob only working locally and communicating classical information. This process, which we call entanglement concentration, has been studied over the past few years and descriptions of local operations to do the job have been given. However, to my knowledge, a study of the difficulty of actually implementing these local operations has not been conducted. For such procedures to ever become practical, efficient algorithms for performing these local operations will be needed. It is this problem that will be the topic of investigation for the 4th-year workshop.

Design methodology

The work in September began with a review of how to implement classical algorithms on a quantum computer (e.g. computing the Hamming weight of a string (this is a key part of the entanglement concentration algorithms). The effort for the remainder of the term was focused on a method to generate symmetric states (assuming infinite precision in intermediate calculations). The basis of symmetric states is a very natural basis to consider when working with identical copies of several partially entangled states. Also, this is a good way to get familiar with arbitrary state generation and arbitrary basis changes. The product of this work has been the design of a quantum network which implements the kth symmetric state on n qubits, where k is supplied to the network as a parameter.

Beginning in January, the technique for generating symmetric states will be generalised to allow the generation of arbitrary states. Again, many papers in the literature describe states required by certain protocols, but do not address how difficult it would be to create these states.
In the second half of January, I will repeat the tasks in October, November and December, but without the assumption of infinite precision. In this case, we will wish to perform these tasks with some given fidelity, and I will calculate the precision with which we need to perform the intermediate calculations.

In February and March I will be in a position to tackle the real goal of the project. I will study closely the local operators used in entanglement concentration algorithms, refine them if possible and design (hopefully efficient) quantum networks for performing them.